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RESEARCH ARTICLE Hexagonal Connectivity Maps for Digital Earth May 21, 2014 International Journal of Digital Earth tJDEguide International Journal of Digital Earth Vol. 00, No. 00, Month 201X, 1{18 RESEARCH ARTICLE Hexagonal Connectivity Maps for Digital Earth Ali Mahdavi Amiria∗, Erika Harrison a, Faramarz Samavati a aDepartment of Computer Science, University of Calgary ; (v1.0 released October 2012) Geospatial data is gathered through a variety of different methods. The integration and handling of such data-sets within a Digital Earth framework are very important in many aspects of science and engineering. One means of addressing these tasks is to use a Discrete Global Grid System and map points of the Earth's surface to cells. An indexing mechanism is needed to access the data and handle data queries within these cells.. In this paper, we present a general hierarchical indexing mechanism for hexagonal cells resulting from the refinement of triangular spherical polyhedra representing the Earth. In this work, we establish a 2D hexagonal coordinate system and diamond-based hierarchies for hexagonal cells that enables efficient determination of hierarchical relationships for various hexagonal refinements, and demonstrate its usefulness in Digital Earth frameworks. Keywords: Digital Earth Framework, Hexagonal Subdivision, Indexing, Triangulation, Hierarchy, Multiresolution. 1. Introduction Digital Earth is a framework for the management and manipulation of geospatial data, spanning a multitude of scientific disciplines (Goodchild 2000). In this frame- work, data is assigned to locations and may be analyzed at multiple resolutions. Each resolution of this framework provides data with a specific level of detail. This multiresolution property is beneficial for efficient vector-data and coverage based data (Wartell et al. 2003). For practical purposes, the finest resolution is usually high enough such that cells with area in square millimeters can be supported. One approach in assigning data to the Earth's positions is to discretize the globe into regions called cells. Cells then represent areas containing geospatial informa- tion associated with a point of interest. Several methods exist to partition the Earth. Latitude/longitude lines or Voronoi cells can be used to partition the Earth into regions of irregular size or shape (Chen et al. 2004, Faust et al. 2000). How- ever, regular cells are often more desirable for a Digital Earth framework as they support efficient algorithms for handling important queries such as containment (which cell includes a point), neighborhood finding, and determining hierarchical relationships between cells (Sahr et al. 2003). Discrete Global Grid Systems (DGGS) provide a representation of the Earth with mostly regular cells (Goodchild 2000, Sahr et al. 2003). The cells of DGGS may be triangular, quadrilateral (squares or diamonds) or hexagonal. Hexagonal ∗Corresponding author. Email: [email protected] 1 May 21, 2014 International Journal of Digital Earth tJDEguide Figure 1.: Icosahedral Snyder Equal Area Aperture 3 Hexagonal Grid (ISEA3H) (PYXIS Innovation 2014). cells are particularly desirable due to their unique characteristics, including uniform definition for adjacency and reduced quantization error (Sahr 2011, Snyder et al. 1999). Figure 1 illustrates the Earth partitioned using hexagonal cells. Six types of hexagonal refinement are mostly used throughout the literature: 1-to-3, 1-to-4, and 1-to-7 refinement in both their centroid-aligned and vertex- aligned variants (Figure 2). Centroid-aligned refinements (c-refinements) produce refined cells sharing centroids with coarse cells while vertex-aligned refinements (v- refinements) generate refined cells with vertices that are shared with the centroid of coarse cells. To generate multiple resolutions with regular spherical cells, a poly- hedron is refined and its resulting faces are projected to the sphere by a spherical projection. Area preserving projections such as Snyder projections are especially preferable as they simplify data analysis on the Earth (Snyder 1992). (a) (b) (c) (d) (e) (f) Figure 2.: The c-refinements and v-refinements (shown in orange and red, respec- tively) for 1-to-3 refinement ((a), (b)), 1-to-4 refinement ((c),(d)), and 1-to-7 re- finement ((e), (f)). To associate information with cells at different levels of refinement, a data struc- ture is required. Quadtrees (Samet 2005, Tobler and Chen 1986) are commonly used to support spatial queries for quad cells; but require many pointers to establish con- nectivity between nodes, which reduces efficiency in high resolution applications with a large data load. To overcome this issue, several indexing methods have been developed for quadtrees. However, quadtrees and their indexing methods cannot be directly applied in the hexagonal case, due to the lack of congruency of hexagons. As a result, an indexing specifically designed for hexagonal cells or an adapting mechanism to benefit from simple congruent shape of quads is required that can efficiently support essential queries. Existing hexagonal indexing methods primarily operate on a complicated fractal- like coverage hierarchy that makes the operations difficult to handle. They are also defined only for a specific type of refinement or polyhedron. In this paper, we introduce a general scheme for indexing hexagonal cells based on modifications of hexagonal coordinate systems. This indexing maintains the hierarchical rela- tionships between successive resolutions. The proposed scheme is general and not 2 May 21, 2014 International Journal of Digital Earth tJDEguide dependent on a specific type of polyhedron or refinement, handling essential queries such as hierarchical traversal between cells and neighborhood finding in constant time. Instead of using fractal coverage hierarchies, we define two simple diamond- based hierarchical coverage for hexagons and demonstrate their usefulness for a Digital Earth framework through several example results. As part of our method evaluation, we compare our indexing with PYXIS indexing (Peterson 2006, Vince and Zheng 2009). PYXIS indexing is from the same category of a set of hierarchical indexing methods designed for hexagonal cells that use a fractal hierarchical coverage (Sahr et al. 2003, Gibson and Lucas 1982). We organize the paper as follows: Related work is presented in Section 2. The terminology of the paper is established in Section 3. Section 4 describes our com- prehensive indexing method. Hierarchy is formally defined in Section 5, and two variations of hexagonal hierarchy as well as their benefits are also presented. In Section 6, we provide comparison and results. We describe a common hierarchical indexing method called PYXIS indexing and compare results with our method. We finally conclude in Section 7. 2. Related Work One way to represent the Earth is to use DGGS. These systems differ from each other based on the underlying polyhedron, type of cells, refinement, projection and data structure employed. In the following, we provide some work related to each of these elements. For a complete survey, please refer to Goodchild (2000) and references therein. Underlying Polyhedra: Numerous polyhedra have been used to approximate the Earth. For instance, an octahedron can be projected to the sphere in such a way that each face corresponds to an octant of the latitude/longitude spherical coordinate system (White 2000). The tetrahedron has been used to represent the Earth in 3D engines that render a virtual globe due to its simplicity (Cozzi and Ring 2011). The truncated icosahedron better approximates the sphere (White et al. 1992). Note that the truncated icosahedron can be constructed by refining the regular icosahedron which itself is widely used as it induces less distortion as compared with other platonic solids (White et al. 1992). Therefore, the truncated icosahedron is also widely used for approximating the Earth (Fekete and Treinish 1990, Sahr 2008, White 2000, PYXIS Innovation 2014, Vince and Zheng 2009). The cube is also an interesting polyhedron for spherical representation due to its adaptability to Cartesian coordinates, hardware devices and existing data struc- tures such as quadtrees (Alborzi and Samet 2000, Mahdavi-Amiri et al. 2013). The dodecahedron, which has 12 pentagonal faces, has been also used for Earth repre- sentation (Wickman et al. 1974). However, since pentagon-to-pentagon refinement has not been defined, they are not a popular choice for hierarchical representation of the Earth. Type of Cells: Different shapes - such as hexagons, quads or triangles - can be used as cells for GDGGS. For instance, Alborzi and Samet (2000) use quadri- lateral faces of a refined cube while Dutton (1999) uses the triangular faces of an octahedron. In this paper, we consider hexagonal cells. As discussed by Sahr (2011), this type of cell is preferred in many applications due to their uniform adjacency, regularity, and support for efficient sampling and smooth subdivision schemes (He and Jia 2005, Kamgar-Parsi et al. 1989, Claes et al. 2002). Although hexagonal cells may be the best choice for sampling the surface of the Earth, they need to be addressed 3 May 21, 2014 International Journal of Digital Earth tJDEguide and rendered efficiently. We use diamonds (unit squares in hexagonal coordinate systems) to efficiently address hexagons and show how to triangulate them for efficient rendering (White 2000). Type of Refinement: Refinements are used to make finer cells based on initial coarse cells. Refinements are widely used in subdivision surfaces to make smooth objects (see Cashman 2012, and references therein). They can also be used to con- struct more cells on the sphere by refining polyhedral faces. To show the generality of our approach, we use six types of hexagonal refinement: 1-to-3, 1-to-4, and 1- to-7 c-refinements and v-refinements (Figure 2). Each of these refinements features some benefits over the others. 1-to-3 refinement increases the number of faces at a lower rate compared to the other two. Under this refinement, more resolutions are produced under a fixed maximum number of faces and, therefore, enables a smoother transition between resolutions. For example, Sahr (2008) uses hexagonal 1-to-3 c-refinement on an icosahedron while Vince (2006) uses the same refinement on the octahedron.
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